General chemistry

Liquids, solids, crystals

How atoms pile up in a crystal, what a unit cell is and how tightly different structures pack.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionIntuition: a repeating box

In a gas the particles are far apart, in a liquid they touch but move past one another, and in a crystalline solid they sit in a regular pattern that repeats in all directions. The smallest box that repeats to build the whole crystal is the unit cell.

Choose a structure, use 2–3 cells per edge to see the repeating pattern, and raise the atom size to 1 to make neighbors touch.

SchoolSchool level: counting atoms in a cell

An atom at a corner is shared by 8 cells and counts $1/8$ ; on an edge $1/4$ ; on a face $1/2$ ; inside, $1$ . This gives the number of atoms per cell $Z$ : simple cubic 1, body-centered cubic (BCC) 2, face-centered cubic (FCC) 4. The coordination number counts the nearest neighbors: 6, 8 and 12.

Three cubic metal structures
Structure	Z	Coordination	Packing
Simple cubic	1	6	52 %
BCC	2	8	68 %
FCC	4	12	74 %

Example: Density of copper

Copper is FCC with edge $a = 361$ pm and $M = 63.55$ g/mol. Find its density and atomic radius.

Solution

$\rho = \dfrac{Z M}{N_A a^3} = \dfrac{4 \times 63.55}{6.022\times10^{23} \times (3.61\times10^{-8})^3} \approx 8.97$ g/cm³ (measured 8.96). Atoms touch along the face diagonal, $4r = \sqrt2\,a$ , so $r = 128$ pm.

UndergraduateUndergraduate: ionic and covalent lattices

In NaCl each ion is surrounded by six of the opposite charge (coordination 6 : 6); in CsCl the larger $\ce{Cs+}$ has eight neighbors (8 : 8). The ratio of ionic radii decides which structure forms. Diamond has each carbon bonded to four others in a tetrahedron, so it is rigid but open, with packing of only 34 %.

\Delta H_{\text{fus}} = T_{\text{fus}} \Delta S_{\text{fus}}

n\lambda = 2d\sin\theta \quad \text{(Bragg)}

P = \frac{nRT}{V - nb} - \frac{an^2}{V^2} \quad \text{(van der Waals)}